Spaces {state space}| can have possible states and an operation on one or more states that makes a state. States are n-dimensional point sets (vectors), as in Hilbert spaces.
metric
Spaces allow measurements. Measurability has rules {convergence rule}.
trajectory
Spaces can have initial states and goal states, which link along intermediate-state paths (trajectory). Distance metric, matching features, or total optimal value heuristic search can find trajectory.
spaces and groups
The possible states are elements, so the state-transition operation makes a group. Transformations represent forces and motions.
matrix
Matrices {second-order tensor} can describe states and operations, which operate on vector states to change state vectors or find vector interaction results. Matrix transformations form groups.
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Date Modified: 2022.0225